An analogue of the Bombieri-Vinogradov theorem for Fourier coefficients of cusp forms
In this work, the author considered the problem of analogues of the Bombieri-Vinogradov Theorem and the Barban-Davenport-Halberstam theorem on primes in arithmetic progressions for Fourier coefficients of cusp forms. For this reason, he set theree functions \[ D_{f}(X;q,a) =\sum_{n\leq X,\ n\equiv a(q)}\lambda_{f}(n)-\frac{1}{\phi (q)}\sum_{n\leq X,\ (n,q)=1}\lambda_{f}(n), \] \[ P_{f}(X;q,a) =\sum_{p\leq X,\ p\equiv a(q)}\lambda_{f}(p)-\frac{1}{\phi (q)}\sum_{p\leq X,\ (p,q)=1}\lambda_{f}(p),\ p\text{ is prime} \] and \[ D_{f}^{\ast}(X;q,a)=\sum_{p\leq X,\ n\equiv a(q)}\lambda_{f}(n)\Lambda (n)-\frac{1}{\phi (q)}\sum_{n\leq X,\ (n,q)=1}\lambda_{f}(p)\Lambda (n) \] and stated an analogue of the classical Bombieri-Vinogradov Theorem, namely, \textbf{Theorem 1.1} Given any \(B>0\), there exists \(C=C(B)>0\) such that \[\sum_{q\leq Q}\max_{(a,q)=1}|D_{f}^{\ast}(X;q,a)|\ll X(\log X)^{-C},\] where \(Q=X^{\frac{1}{2}}(\log X)^{-B}\). Also he stated an analogous to the classical Barban-Davenport-Halberstam Theorem, namely, \textbf{Theorem 1.2} Given any \(B>0\), there exists a constant \(C(B)>0\) such that \[ \sum_{q\leq Q}\sum_{(a,q)=1}|P_{f}(X;q,a)|^{2}\ll X^{2}\exp (-B(\log X)^{\frac{1}{2}}), \] if \(Q=X\exp (-C(\log X)^{\frac{1}{2}})\). Further he obtained the following theorem which treated the case where the inner sums are not restricted over primes. \textbf{Theorem 1.3} For every \(\varepsilon >0\) and some \(C>0\), \[ \sum_{q\leq Q}\sum_{(a,q)=1}|D_{f}(X;q,a)|^{2}\ll X(Q+X^{\frac{11}{12}+\varepsilon})(\log Q)^{C}. \] It is clear to replace the divisor function by Fourier coefficients of a fixed cusp form \(f\) and one can \[ S(X):=\sum_{n\leq X}\Lambda (n)\lambda_{f}(n-a), \] where \(a\) is any integer. Also set \[ S_{0}(X):=\sum_{p\leq X}\lambda_{f}(p)\tau (p-1). \] Then from Theorem 1.1 he obtained that \textbf{Theorem 1.4} For the set \(S_{0}(X)\), \[S_{0}(X)\ll \frac{X\log (\log X)}{\sqrt{\log X}},\] where the implied constant depends only on \(f\).
scientific article; zbMATH DE number 6851615